Laplace transforms:Mass-Spring Oscillator

Problem Statement:

An ideal mass m sliding on a frictionless surface, attached via an ideal spring k to a rigid wall. The spring is at rest when the mass is centered at x=0. Find the equation of motion that the spring mass follows.

Solution:

By Newton's first law:

$\mathbf{F}=ma\Rightarrow {\mathbf{f}}_m(t)=m\ddot{x}$

By Hooke's law:

$\mathbf{F}=kx\Rightarrow {\mathbf{f}}_k(t)=mx$

By Newton's third law of motion that states every action produces an equal and opposite reaction, we have f_k = -f_m. That is, the force f_k applied by the mass to the spring is equal and opposite to the accelerating force f_m exerted in the -x direction by the spring on the mass.

${\mathbf{f}}_m(t)+{\mathbf{f}}_k(t)=0\Rightarrow m\ddot{x}(t)+kx(t)=0$

We now have a second order differential equation that governs the motion of the mass. Taking the Laplace transform of both sides gives:

<math>0=\mathcal{L}_s\left\{m\ddot{x}(t)+k{x}(t)\right\}